I wouldn't bother going to MyMy, really. If you do go, their finger-sized meat pies are about the only edible thing.
Of all the places referenced in the link above, Barashka is the only really sensible suggestion. Simple Pleasures used to be good but not any longer. The rest... well, I'd stay away
A couple more places you might fnd interesting are Le Carré Blanc and Correa's. Neither is particularly Russian, though. Correa's would be less expensive of the two. They have several branches around town. Their website is new.correas.ru
If you want homey Russian cooking, Ryumka is a good, moderately-priced place on Tverskaya, the city's main drag. (I haven't been there for a year or so but they used to be consistently good, year after year). It's basically a vodka place but the food is (or used to be) just right.
By all means try some ethnic cooking you can't get elsewhere, like Ukrainian (Shinok is nice but a bit on the expensive side), Georgian, Uzbek or Armenian. The best Armenian place in town is in the local Hyatt but you'll pay through the nose. Noah's Ark is way more affordable - if you take it easy on the liquor and wine - and still great.
Red Square, 1 (see everbrite's reply above) is supposed to be good but I haven't been there. The food they serve is a recreation of historical recipes from the Czars' table.
Hope it helps
